Efficient Algorithms for Attributed Graph Alignment with Vanishing Edge Correlation

TL;DR

This paper introduces a polynomial-time algorithm for graph alignment that leverages attribute information to achieve exact vertex correspondence even when edge correlation vanishes as the graph size grows.

Contribution

It demonstrates that with minimal attribute data, exact graph alignment is feasible in polynomial time under vanishing edge correlation, challenging previous conjectures.

Findings

Exact recovery achieved with vanishing edge correlation using attribute info

Proposed local tree structure and subgraph counting technique

Algorithm recovers most vertices then refines for full accuracy

Abstract

Graph alignment refers to the task of finding the vertex correspondence between two correlated graphs of $n$ vertices. Extensive study has been done on polynomial-time algorithms for the graph alignment problem under the Erd\H{o}s-R\'enyi graph pair model, where the two graphs are Erd\H{o}s-R\'enyi graphs with edge probability $q_\mathrm{u}$, correlated under certain vertex correspondence. To achieve exact recovery of the correspondence, all existing algorithms at least require the edge correlation coefficient $\rho_\mathrm{u}$ between the two graphs to be \emph{non-vanishing} as $n\rightarrow\infty$. Moreover, it is conjectured that no polynomial-time algorithm can achieve exact recovery under vanishing edge correlation $\rho_\mathrm{u}<1/\mathrm{polylog}(n)$. In this paper, we show that with a vanishing amount of additional \emph{attribute information}, exact recovery is polynomial-time feasible under \emph{vanishing} edge correlation $\rho_\mathrm{u} \ge n^{-\Theta(1)}$. We identify a \emph{local} tree structure, which incorporates one layer of user information and one layer of attribute information, and apply the subgraph counting technique to such structures. A polynomial-time algorithm is proposed that recovers the vertex correspondence for most of the vertices, and then refines the output to achieve exact recovery. The consideration of attribute information is motivated by real-world applications like LinkedIn and Twitter, where user attributes like birthplace and education background can aid alignment.